A survey of simplicial, relative, and chain complex homology theories for hypergraphs

TL;DR

This survey reviews recent methods for constructing homology theories from hypergraphs using simplicial, relative, and chain complexes, highlighting their properties, functoriality, and applications through examples.

Contribution

It introduces nine different homology constructions for hypergraphs, analyzing their properties and demonstrating their use with illustrative examples.

Findings

Multiple homology theories capture various hypergraph properties

Functoriality is established for several homology theories

Examples illustrate the variability and applicability of the methods

Abstract

Hypergraphs have seen widespread application in network and data science communities in recent years. We present a survey of recent work to construct auxiliary structures from hypergraphs -- specifically simplicial, relative, and chain complexes -- that can be used to build homology theories for hypergraphs. We define and describe nine different constructions and their associated homology theories. We discuss some interesting properties of each homology theory to show how various hypergraph properties imply properties of the homology groups. We also include discussion of functoriality for several of the homology theories. Finally, we provide a series of illustrative examples by computing many of these homology theories for small hypergraphs to show the variability of the methods and build intuition.